Consider the given grammar A → eS | Cr| ξ C → f/p/ξ
S → Abc | ad
Compute FIRST (A)
FIRST (A) = {e, f, p, r, \(\epsilon\) }
Understanding the FIRST Set Computation
This solution explains the process of calculating the FIRST set for a specific non-terminal 'A' using the rules of a given context-free grammar. The FIRST set of a non-terminal contains all the possible terminal symbols that can begin a string derived from that non-terminal, including the empty string (\(\epsilon\)) if the non-terminal can derive it.
Grammar Definition
The context-free grammar provided is:
- S → Abc | ad
- A → eS | Cr | ξ
- C → f | p | ξ
In this grammar, 'ξ' represents the empty string (epsilon, denoted as \(\epsilon\)). 'S', 'A', and 'C' are non-terminal symbols, while 'a', 'b', 'c', 'e', 'f', 'p', 'r', and 'd' are terminal symbols.
Computing FIRST(A) Step-by-Step
We need to find the set of all terminals that can start a string derived from 'A'. We examine each production rule for 'A':
Step 1: Analyzing Production A → eS
The rule states that 'A' can directly produce 'e' followed by 'S'. Since 'e' is a terminal symbol, it is the first possible symbol in a string derived from 'A'.
Therefore, 'e' is added to FIRST(A).
Contribution from this rule: {e}
Step 2: Analyzing Production A → Cr
This rule states that 'A' can produce 'C' followed by 'r'. Here, 'C' is a non-terminal. To find the terminals that 'A' can start with via this rule, we first need to find the FIRST set of C (FIRST(C)).
Calculating FIRST(C)
Let's look at the rules defining 'C':
- C → f: 'f' is a terminal. So, 'f' ∈ FIRST(C).
- C → p: 'p' is a terminal. So, 'p' ∈ FIRST(C).
- C → ξ: This rule allows 'C' to derive the empty string, \(\epsilon\). So, \(\epsilon\) ∈ FIRST(C).
Combining these, we get: FIRST(C) = {f, p, \(\epsilon\)}.
Applying FIRST(C) to A → Cr
Now, we use the calculated FIRST(C) for the rule A → Cr:
- The terminals 'f' and 'p' from FIRST(C) are added to FIRST(A).
- Crucially, FIRST(C) includes \(\epsilon\). This means 'C' can disappear, and the derivation continues with the *next* symbol in the rule A → Cr, which is 'r'. Since 'r' is a terminal, it is also added to FIRST(A).
Contribution from this rule: {f, p, r}
Step 3: Analyzing Production A → ξ
This rule directly shows that 'A' can derive the empty string, \(\epsilon\).
Therefore, \(\epsilon\) is added to FIRST(A).
Contribution from this rule: {\(\epsilon\)}
Combining the Results for FIRST(A)
To get the complete FIRST(A) set, we unite the contributions from all the rules for 'A':
- From A → eS: {e}
- From A → Cr: {f, p, r}
- From A → ξ: {\(\epsilon\)}
Combining these sets: FIRST(A) = {e} \(\cup\) {f, p, r} \(\cup\) {\(\epsilon\)}
This results in: FIRST(A) = {e, f, p, r, \(\epsilon\)}.
Final Answer
The computed FIRST set for the non-terminal A is {e, f, p, r, \(\epsilon\)}.
Important Questions from Context Free Grammars
- Which of the following Grammars is/are only Context Free?
A S $\rightarrow$ Ab
aS$\rightarrow$ aA
A$\rightarrow$ aB S $\rightarrow$ Ab
A $\rightarrow$ Sa
A $\rightarrow$ aC S $\rightarrow$ AS
S $\rightarrow$ aA
A$\rightarrow$ aD S $\rightarrow$ Ab
S $\rightarrow$ aA
A$\rightarrow$ aE S $\rightarrow$ Sb
A $\rightarrow$ Aa
A $\rightarrow$ $\varepsilon$
Choose the correct answer from the options given below: Which of the following grammars is (are)ambiguous ?
(1) S → SS | aSb | bSa |e(2) S → aSbS | bSaS |e
(3) S → aAB
A → bBb
B → A Ι eConsider the grammar,
G = S → abScB | e,
B→ bB | b.
What is the language generated by the given grammar ?
Select the sentence which is meaningful and grammatically correct.
- Consider the following context-free grammar $G$.
$$S \rightarrow abaABabba$$
$$A \rightarrow aaBBAB \mid bBabaa$$
$$B \rightarrow aBb \mid ab$$
In the above grammar, $S$ is the start symbol, $a$ and $b$ are terminal symbols, and $A$ and $B$ are non-terminal symbols.
Let $L(G)$ be the language generated by the grammar $G$. For a string $s \in L(G)$, let $n_1(s)$ be the number of $a$’s in $s$ and $n_2(s)$ be the number of $b$’s in $s$.
Which of the following statements is/are true?
